Collision kernels and laser spectroscopy

Abstract

Collisional processes occurring within an atomic vapor can be conveniently described in terms of collision kernels. The population kernel $W_{\mathrm{ii}}({\overrightarrow{\mathrm{v}}}^{\prime }\to \overrightarrow{\mathrm{v}})$ gives the probability density per unit time that an "active" atom in state $i$ undergoes a collision with a perturber that changes the active atom's velocity from ${\overrightarrow{\mathrm{v}}}^{\prime }$ to $\overrightarrow{\mathrm{v}}$. For active atoms in a linear superposition of states $i$ and $j$, there is an analogous coherence kernel $W_{\mathrm{ij}}({\overrightarrow{\mathrm{v}}}^{\prime }\to \overrightarrow{\mathrm{v}}) \left(i\ne j\right)$ reflecting the effects of collisions on the off-diagonal density-matrix element ${\rho }_{\mathrm{ij}}$. In this work, we discuss the general properties of the collision kernels which characterize a two-level active atom which, owing to the action of a radiation field, is in a linear superposition of its two levels. Using arguments based on the uncertainty principle, we show that collisions can be divided roughly into the following two categories: (1) collisions having impact parameters less than some characteristic radius which may be described classically and (2) collisions having impact parameters larger than this characteristic radius which give rise to diffractive scattering and must be treated using a quantum-mechanical theory. For the population kernels, collisions of type (1) can lead to a large-angle scattering component, while those of type (2) lead to a small-angle (diffractive) scattering component. For the coherence kernel, however, assuming that the collisional interaction for states $i$ and $j$ differ appreciably, only collisions of type (2) contribute, and the coherence kernel contains a small-angle scattering component only. The absence of a large-angle scattering component in the coherence kernel is linked to a collision-induced spatial separation of the trajectories associated with states $i$ and $j$. Interestingly enough, the width of the diffractive kernel, as measured in the laboratory frame, is found to be insensitive to the perturber to active-atom mass ratio. To illustrate these features, a specific calculation of the kernels is carried out using a hard-sphere model for the scattering. The relationship of the present description of collisions to that of traditional pressure-broadening theory in which trajectory separation effects are ignored is discussed. It is explained why traditional pressure-broadening theory correctly describes collision effects in linear spectroscopy, but fails to provide an adequate description of some saturation spectroscopy and photonecho experiments in which velocity-changing collisions associated with the coherence kernel play a significant role. An expression for the collisionally modified photon-echo amplitude is derived which clearly displays the role played by velocity-changing collisions associated with the coherence kernel.